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A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Published online by Cambridge University Press:  28 May 2015

Jan Brandts  and
Ricardo R. da Silva
Jan Brandts*
Affiliation:
University of Amsterdam, Faculty of Science, Korteweg-de Vries Institute for Mathematics, Science Park 904, Amsterdam, Netherlands
Ricardo R. da Silva*
Affiliation:
University of Amsterdam, Faculty of Science, Korteweg-de Vries Institute for Mathematics, Science Park 904, Amsterdam, Netherlands
*
Corresponding author. Email: J.H.Brandts@uva.nl
Corresponding author. Email: R.J.ReisdaSilva@uva.nl
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Abstract

Given two n × n matrices A and A0 and a sequence of subspaces with dim the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk(A0 − A)Πk, where Πk is the orthogonal projection on . Consequently, Akν = Aν and ν*Ak = ν*A for all Thus is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Akxk = b are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests.

Keywords

65F1065F08Linear systemGalerkin methodsubspace projected approximate matrixminimal residual method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 2 , May 2013 , pp. 120 - 137
Copyright
Copyright © Global-Science Press 2013

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References

[1]Beattie, C. (1998). Harmonic Ritz and Lehmann bounds. Electron. Trans. Numer. Anal., 7:18–39.Google Scholar
[2]Brandts, J.H. (2003). The Riccati Method for eigenvalues and invariant subspaces of matrices with inexpensive action. LinearAlgebra Appl., 358:335–365.Google Scholar
[3]Brandts, J.H. and da Silva, R.R. (2011). On the subspace projected approximate matrix method. arXiv:1103.1779v1 [math.NA] (23 pages)Google Scholar
[4]Chen, W. and Poirier, B. (2006). Parallel implementation of efficient preconditioned linear solver for grid-based applications in chemical physics. II J. Comput. Phys., 219:198–209.CrossRefGoogle Scholar
[5]Kaczmarz, S. (1993). Approximate solution of systems of linear equations. Translated from the German. Internat. J. Control 57,6:1269–1271.Google Scholar
[6]Lanczos, C. (1950). An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Research Nat. Bur. Standards, 45(4):255–282.Google Scholar
[7]Golub, G.H. and van Loan, C.F. (1996). MatrixComputations(third edition). The John Hopkins University Press, Baltimore and London.Google Scholar
[8]Györffy, W., Seidler, P. and Christiansen, O. (2009). Solving the eigenvalue equations of correlated vibrational structure methods: Preconditioning and targeting strategies. J. Chem. Phys., 131:024108 (16 pages)Google Scholar
[9]Hestenes, M.R. and Stiefel, E. (1954). Methods of conjugate gradients for solving linear systems. J.Res. Nat.Bur. Stand., 49:409–436.Google Scholar
[10]Jia, Z. and Stewart, G.W. (2001). An analysis of the Rayleigh-Ritz method for approximating eigenspaces. Math. Comp., 70(234):637–647.Google Scholar
[11]Medvedev, D. M., Gray, S. K., Wagner, A. F., Minkoff, M. and Shepard, R. (2005). Advanced software for the calculation of thermochemistry, kinetics, and dynamics. J. Physics: Conference Series, 16:247.Google Scholar
[12]Paige, C.C. and Saunders, M.S. (1975). Solution of sparse indefinite systems of linear equations. SIAMJ. Numer. Anal., 2:617–629.Google Scholar
[13]Parlett, B.N. (1998). The Symmetric Eigenvalue Problem. SIAM Classics in Applied Mathematics 20, Philadelphia.Google Scholar
[14]Ribeiro, F., Lung, C. and Leforestier, C. (2005). A Jacobi-Wilson description coupled to a block-Davidson algorithm: An efficient scheme to calculate highly excited vibrational levels. J. Chem. Phys., 123:054106 (10 pages)CrossRefGoogle ScholarPubMed
[15]Saad, Y. (1981). Krylov subspace method for solving large unsymmetric linear systems. Math. Comp., 37:105–126.Google Scholar
[16]Saad, Y. and Schultz, M.H. (1986). GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856–869.Google Scholar
[17]Shepard, R., Wagner, A.F., Tilson, J.L. and Minkoff, M. (2001). The subspace-projected approximate matrix (SPAM) modification of the Davidson method. J. Comput. Phys., 172(2):472–514.CrossRefGoogle Scholar
[18]Sleijpen, G.L.G., van den Eshof, J. and Smit, P. (2003). Optimal a priori error bounds for the Rayleigh-Ritz method. Math.Comp., 72(242):677–684.Google Scholar
[19]Sleijpen, G.L.G. and van der Vorst, H.A. (1996). Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Applic., 17:401–425.CrossRefGoogle Scholar
[20]Zhou, Y., Shepard, R. and Minkoff, M. (2005). Computing eigenvalue bounds for iterative subspace matrix methods. Comput. Phys. Commun., 167:90–102.Google Scholar